{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

integral

# integral - COMMUTATIVE ALGEBRA PETE L CLARK Contents...

This preview shows pages 1–4. Sign up to view the full content.

COMMUTATIVE ALGEBRA PETE L. CLARK Contents Introduction 4 0.1. What is Commutative Algebra? 4 0.2. Why study Commutative Algebra? 4 0.3. Some themes of Commutative Algebra 7 1. Commutative rings 7 1.1. Fixing terminology 7 1.2. Adjoining elements 10 1.3. Ideals and quotient rings 11 1.4. Products of rings 14 1.5. A cheatsheet 16 2. Galois connections 17 2.1. The basics 17 2.2. Examples of Antitone Galois Connections 18 2.3. Examples of Isotone Galois Connections 20 3. Modules 20 3.1. Basic definitions 20 3.2. Finitely presented modules 25 3.3. Torsion and torsionfree modules 28 3.4. Tensor and Hom 28 3.5. Projective modules 30 3.6. Injective modules 37 3.7. Flat modules 45 3.8. Nakayama’s Lemma 46 3.9. Ordinal Filtrations and Applications 50 3.10. Tor and Ext 57 3.11. More on ﬂat modules 65 3.12. Faithful ﬂatness 71 4. First Properties of Ideals in a Commutative Ring 74 4.1. Introducing maximal and prime ideals 74 4.2. Radicals 77 4.3. Comaximal ideals 79 4.4. Local rings 82 5. Examples of rings 83 5.1. Rings of numbers 83 5.2. Rings of continuous functions 84 5.3. Rings of holomorphic functions 90 5.4. Polynomial rings 92 5.5. Semigroup algebras 94 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 PETE L. CLARK 6. Swan’s Theorem 100 6.1. Introduction to (topological) vector bundles 101 6.2. Swan’s Theorem 102 6.3. Proof of Swan’s Theorem 103 6.4. Applications of Swan’s Theorem 106 7. Localization 109 7.1. Definition and first properties 109 7.2. Pushing and pulling via a localization map 111 7.3. The fibers of a morphism 113 7.4. Commutativity of localization and passage to a quotient 114 7.5. Localization at a prime ideal 114 7.6. Localization of modules 115 7.7. Local properties 116 7.8. Local characterization of finitely generated projective modules 118 8. Noetherian rings 121 8.1. Chain conditions on posets 122 8.2. Chain conditions on modules 122 8.3. Semisimple modules and rings 123 8.4. Normal Series 126 8.5. The Krull-Schmidt Theorem 128 8.6. Some important terminology 132 8.7. Introducing Noetherian rings 133 8.8. The Bass-Papp Theorem 135 8.9. Artinian rings: structure theory 136 8.10. The Hilbert Basis Theorem 139 8.11. The Krull Intersection Theorem 140 8.12. Krull’s Principal Ideal Theorem 143 8.13. The Dimension Theorem 145 8.14. The Artin-Tate Lemma 146 9. Boolean rings 146 9.1. Definition and first properties of Boolean rings 146 9.2. Boolean Algebras 147 9.3. Ideal Theory in Boolean Rings 150 9.4. The Stone Representation Theorem 152 9.5. Boolean spaces 153 9.6. Stone Duality 156 9.7. Topology of Boolean Rings 156 10. Primary Decomposition 157 10.1. Some preliminaries on primary ideals 157 10.2. Primary decomposition, Lasker and Noether 160 10.3. Irredundant primary decompositions 161 10.4. Uniqueness properties of primary decomposition 162 10.5. Applications in dimension zero 165 10.6. Applications in dimension one 165 11. Aﬃne k -algebras and the Nullstellensatz 165 11.1. Zariski’s Lemma 166 11.2. Hilbert’s Nullstellensatz 167 11.3. Other Nullstellens¨atze 170
COMMUTATIVE ALGEBRA 3 12. Goldman domains and Hilbert-Jacobson rings 170 12.1. Goldman domains 171 12.2. Hilbert rings 174 12.3. Jacobson Rings 175 12.4. Hilbert-Jacobson Rings 175 13. The spectrum of a ring 176 13.1. The Zariski spectrum 176 13.2. Properties of the spectrum: quasi-compactness 177 13.3. Properties of the spectrum: separation and specialization 178 13.4. Irreducible spaces 180 13.5. Noetherian spaces 182 13.6.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 264

integral - COMMUTATIVE ALGEBRA PETE L CLARK Contents...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online